Consider the set $\mathcal S(p)$
of symmetric matrices $A$ of size $p\times p$
with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$.


Let $\alpha>0$ be fixed.
For a matrix $A$ as above, let $J_p^\alpha(A) = \{j=1,\dotsc,p: \lVert A e_j\rVert_1 \ge p^\alpha\}$
be the indices of the columns with "large" $\ell_1$ norm.
- How fast does
$$u_p^\alpha = \sup_{A\in \mathcal S(p)} \lvert J_p^\alpha(A)\rvert$$ grow as $p\to\infty$?
For any value of $\alpha$,
both non-trivial upper and lower bounds would be interesting.
- Are there some phase transition with respect to $\alpha$, e.g., some $\alpha_*$ such that $u_p^\alpha$ grows very differently for $\alpha>\alpha_*$ and $\alpha<\alpha_*$?

Of course, one has the bound for $A$ with non-negative entries
$p^\alpha \lvert J\rvert  \le \sum_{j\in J} \lVert c_j\rVert_1\le \sum_{ij}\lvert a_{ij}\rvert=\sum_{ij} a_{ij} \le p \lVert A\rVert_\text{op}$, but I suspect that the general case will be much different.